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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeApr 8th 2010
• (edited Aug 17th 2011)

More at field with one element, after creating person entry Christophe Soulé about the creator. By the way the Soulé has different encoding in n-Forum than in nlab so the link does not access the right page from here. See redirect Christophe Soule.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeApr 8th 2010

New related entry Riemann hypothesis. I plan to have Riemann zeta function separetely from zeta function as the latter may refer to much wider range of zeta functions of dynamical systems, L-functions of varieties, various arithmetic zeta functions, zeta functions in the operator theory and so on…all of which have some related ideas.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeApr 8th 2010

Created number theory. I crudely allude to Matijasevich’s theorem; foundational experts whose presence in $n$lab is widely felt are invited to improve it.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeApr 8th 2010

Added some possibly controversial remarks to discrete mathematics refering to number theory.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeAug 17th 2011

More references (including a short video of Connes) at field with one element.

• CommentRowNumber6.
• CommentAuthorfpaugam
• CommentTimeAug 18th 2011
Modified the beginning of the entry by adding details on the origins of the geometry over F1, that are really due to Tits, and not to Soulé. Soulé reintroduced these with more zeta functions in mind but Tits already had a great deal of geometric ideas in his work on combinatorial geometry (that could be called geometry over F1) and algebraic groups vs. Weyl groups.
• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeMay 11th 2012

I added redirects absolute algebraic geometry and absolute geometry. Though it is in principle different to talk about geometry over $\mathbf{F}_1$ and on $\mathbf{F}_1$, in practice it is always convoluted so it is better not to artifically split. New entry blueprint with redirect blue scheme.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 11th 2012
• (edited May 11th 2012)

I have edited the very first sentence at field with one element in an attempt to make it be clearer: now it reads:

Jacques Tits first observed that many identities coming from algebraic geometry (and particularly algebraic groups) over finite fields $\mathbb{F}_q$ made perfect sense as expressions in $q$ when extrapolated to the case $q=1$, even though, of course, there is no actual field with a single element.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeMay 11th 2012
• (edited May 11th 2012)

Something essential disappeared in your version:

and even correspond though, to interesting combinatorial results

If I understood right, it is not merely that one can DEFINE the extensions to $q=1$, but the values for $q=1$ were in correlation with various interesting combinatorial (i.e. counting) (and representation theoretical) facts in special cases. Though I do not know much about original Tits’ work.

More references at blueprint.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 11th 2012

Sure, right. Sorry if I deleted that. As soon as the $n$Lab responds again (it is being very unresponsive these days…) I’ll clarify.

Also, I would prefer if the entry does not start with the words “Jacques Tits first…”. I think we should first give the facts, and then the history. Nothing against Jacques Tits, we can honor him greatly a few paragraphs below, but I think it is better style not to mix explanation of a subject with origin of a subject.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMay 11th 2012

Okay, I have tried to polish the first two paragraphs a bit more. Please check if you can live with it.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeMay 11th 2012

I agree with you in general. But as long as we did not put the contemporary insight into contemporary motivation it was in a sense better as Tits in the first sentence. Namely as you put it now, it looks like the combinatorial identities of Tits are THE motivation for the subject, not only that there were in the beginnings. So this motivation in the first paragraph is related to an early historical phase. Now Riemann conjecture, tropical geometry, lambda rings etc. tell much more different motivation and related phenomena of stronger geometric nature, than the series in earlier works. I might be wrong. I am not suggesting that you reverse back to Tits in the first sentence but that you are aware that this is temporary state (I should think more in order to improve) and that it does mislead a bit. From next week I am at IHES for 4 weeks and hopefully I will be able to work more in nLab, though I have some urgent papers to finish and some other things as well.